Final answer:
The highest hardness in copper is generally observed with nanometer-sized grains. Smaller grain sizes like 50 nanometers increase hardness due to the grain boundary strengthening effect.
Step-by-step explanation:
The volume of copper particles at which hardness is the highest is related to the Hall-Petch relationship, which suggests that as grain size decreases, the hardness and yield strength of the material increase. While the specific value for the highest hardness in copper may vary based on experimental conditions and the purity of the copper, generally, smaller grain sizes in the range of nanometers often result in higher hardness compared to micrometer-sized grains. In materials science, it is well-known that when the grain size reaches the nanometer scale, such as 50 nanometers or below, the hardness of metallic materials often increases substantially due to the grain boundary strengthening effect.